reserve x, y, z for set,
  T for TopStruct,
  A for SubSpace of T,
  P, Q for Subset of T;

theorem
  for T being non empty TopSpace holds T is compact iff for F being
  Subset-Family of T st F <> {} & F is closed & meet F = {} ex G being
  Subset-Family of T st G <> {} & G c= F & G is finite & meet G = {}
proof
  let T be non empty TopSpace;
  thus T is compact implies for F being Subset-Family of T st F <> {} & F is
  closed & meet F = {} ex G being Subset-Family of T st G <> {} & G c= F & G is
  finite & meet G = {}
  proof
    assume
A1: T is compact;
    let F be Subset-Family of T such that
A2: F <> {} and
A3: F is closed and
A4: meet F = {};
    not F is centered by A1,A3,A4,Th4;
    then consider G being set such that
A5: G <> {} and
A6: G c= F and
A7: G is finite and
A8: meet G = {} by A2,FINSET_1:def 3;
    reconsider G as Subset-Family of T by A6,XBOOLE_1:1;
    take G;
    thus thesis by A5,A6,A7,A8;
  end;
  assume
A9: for F being Subset-Family of T st F <> {} & F is closed & meet F =
  {} ex G being Subset-Family of T st G <> {} & G c= F & G is finite & meet G =
  {};
  for F being Subset-Family of T st F is centered & F is closed holds
  meet F <> {}
  proof
    let F be Subset-Family of T;
    assume that
A10: F is centered and
A11: F is closed;
    assume
A12: meet F = {};
    F <> {} by A10,FINSET_1:def 3;
    then ex G being Subset-Family of T st G <> {} & G c= F & G is finite &
    meet G = {} by A9,A11,A12;
    hence contradiction by A10,FINSET_1:def 3;
  end;
  hence thesis by Th4;
end;
